1.1.1 Poisson structures and symplectic groupoids

Recall that a symplectic groupoid is a pair $(\mathcal {G},\Omega )$ consisting of a Lie groupoid $\mathcal {G}$ and a symplectic form $\Omega$ on $\mathcal {G}$ which is multiplicative. That is, it is compatible with the groupoid structure in the sense that

\[ ({\rm pr}_1)^*\Omega=m^*\Omega-({\rm pr}_2)^*\Omega, \]

where we use

\[ m,{\rm pr}_1,{\rm pr}_2:\mathcal{G}^{(2)}\to \mathcal{G} \]

to denote the groupoid multiplication and the projections from the space of composable arrows $\mathcal {G}^{(2)}$ to $\mathcal {G}$. Given a symplectic groupoid $(\mathcal {G},\Omega )\rightrightarrows M$, there is a unique Poisson structure $\pi$ on $M$ with the property that the target map $t:(\mathcal {G},\Omega )\to (M,\pi )$ is a Poisson map. The Lie algebroid of $\mathcal {G}$ is canonically isomorphic to the Lie algebroid $T^*_\pi M$ of the Poisson structure $\pi$ on $M$, via

(4)\begin{equation} \rho_\Omega:T^*_\pi M\to \text{Lie}(\mathcal{G}),\quad \iota_{\rho_\Omega(\alpha)}\Omega_{1_x}=({\rm d} t_{1_x})^*\alpha,\quad \forall\alpha\in T^*_xM,\enspace x\in M. \end{equation}

The symplectic groupoid $(\mathcal {G},\Omega )$ is said to integrate the Poisson structure $\pi$ on $M$.

1.1.2 Momentum maps and Hamiltonian actions

To begin with, recall the following definition.

Definition 1.3 [Reference Mikami and WeinsteinMW88]

Let $(S,\omega )$ be a symplectic manifold. A left action of a symplectic groupoid $(\mathcal {G},\Omega )\rightrightarrows M$ along a map $J:(S,\omega )\to M$ is called Hamiltonian if it satisfies the multiplicativity condition

(5)\begin{equation} ({\rm pr}_\mathcal{G})^*\Omega=(m_S)^*\omega-({\rm pr}_S)^*\omega, \end{equation}

where we use

\[ m_S,{\rm pr}_S:\mathcal{G}\ltimes S\to S,\quad {\rm pr}_\mathcal{G}:\mathcal{G}\ltimes S\to \mathcal{G}, \]

to denote the map defining the action and the projections from the action groupoid to $S$ and $\mathcal {G}$. Right Hamiltonian actions are defined similarly.

The infinitesimal version of Hamiltonian actions for symplectic groupoids are momentum maps. To be more precise, by a momentum map we mean a Poisson map $J:(S,\omega )\to (M,\pi )$ from a symplectic manifold into a Poisson manifold. That is, for all $f,g\in C^\infty (M)$ it holds that

\[ J^*\{f,g\}_\pi=\{J^*f,J^*g\}_\omega. \]

Every momentum map comes with a symmetry, in the form of a Lie algebroid action. Indeed, a momentum map $J:(S,\omega )\to (M,\pi )$ is acted on by the Lie algebroid $T^*_\pi M$ of the Poisson structure $\pi$. Explicitly, the Lie algebroid action $a_J:\Omega ^1(M)\to \mathcal {X}(S)$ along $J$ is determined by the momentum map condition:

(6)\begin{equation} \iota_{a_J(\alpha)}\omega=J^*\alpha,\quad \forall \alpha\in \Omega^1(M). \end{equation}

Hamiltonian actions integrate such Lie algebroid actions, in the following sense.

Proposition 1.4 Let $(\mathcal {G},\Omega )\rightrightarrows M$ be a symplectic groupoid and let $\pi$ be the induced Poisson structure on $M$ (as in § 1.1.1). Suppose that we are given a left Hamiltonian $(\mathcal {G},\Omega )$-action along $J:(S,\omega )\to M$. Then $J:(S,\omega )\to (M,\pi )$ is a momentum map and the Lie algebroid action

(7)\begin{equation} a:\Omega^1(M)\to \mathcal{X}(S) \end{equation}

associated to the Lie groupoid action (via (4)) coincides with the canonical $T^*_\pi M$-action along $J$. In other words, (7) satisfies the momentum map condition (6). A similar statement holds for right Hamiltonian actions.

An appropriate converse to this statement holds as well; see, for instance, [Reference Bursztyn and CrainicBC05].

1.2.1 The leaves and normal representations of Lie and symplectic groupoids

To start with, we introduce some more terminology. Let $\mathcal {G}\rightrightarrows M$ be a Lie groupoid and $x\in M$. By the leaf of $\mathcal {G}$ through $x$ we mean the set $\mathcal {L}_x$ consisting of points in $M$ that are the target of an arrow starting at $x$. By the isotropy group of $\mathcal {G}$ at $x$ we mean the group $\mathcal {G}_x:=s^{-1}(x)\cap t^{-1}(x)$ consisting of arrows that start and end at $x$. In general, $\mathcal {G}_x$ is a submanifold of $\mathcal {G}$ and as such it is a Lie group. The leaf $\mathcal {L}_x$ is an initial submanifold of $M$, with smooth manifold structure determined by the fact that

(8)\begin{equation} t:s^{-1}(x)\to \mathcal{L}_x \end{equation}

is a (right) principal $\mathcal {G}_x$-bundle. Note that a leaf of $\mathcal {G}$ may be disconnected. Given a leaf $\mathcal {L}\subset M$ of $\mathcal {G}$, we let $\mathcal {G}_\mathcal {L}:=s^{-1}(\mathcal {L})$ denote the restriction of $\mathcal {G}$ to $\mathcal {L}$. This is a Lie subgroupoid of $\mathcal {G}$. In all of our main theorems, we assume at least that $\mathcal {G}$ is proper at points in the leaves under consideration, in the following sense.

Definition 1.7 [Reference Crainic and StruchinerCS13]

A Hausdorff Lie groupoid $\mathcal {G}$ is called proper at $x\in M$ if the map

\[ (t,s):\mathcal{G}\to M\times M \]

is proper at $(x,x)$, meaning that any sequence $(g_n)$ in $\mathcal {G}$ such that $(t(g_n),s(g_n))$ converges to $(x,x)$ admits a convergent subsequence.

If $\mathcal {G}$ is proper at some (or equivalently every) point $x\in \mathcal {L}$, then $\mathcal {L}$ and the Lie subgroupoid $\mathcal {G}_\mathcal {L}$ are embedded submanifolds of $M$ and $\mathcal {G}$ respectively, and the isotropy group $\mathcal {G}_x$ is compact. Returning to a general leaf $\mathcal {L}$, the normal bundle $\mathcal {N}_\mathcal {L}$ to the leaf in $M$ is naturally a representation

\[ \mathcal{N}_\mathcal{L}\in {\rm Rep}(\mathcal{G}_\mathcal{L}) \]

of $\mathcal {G}_\mathcal {L}$, with the action defined as

(9)\begin{equation} g\cdot[v]=[{\rm d} t(\hat{v})]\in \mathcal{N}_{t(g)},\quad g\in \mathcal{G}_\mathcal{L},\enspace [v]\in \mathcal{N}_{s(g)}, \end{equation}

where $\hat {v}\in T_g\mathcal {G}$ is any tangent vector satisfying ${\rm d} s(\hat {v})=v$. We call this the normal representation of $\mathcal {G}$ at $\mathcal {L}$. It encodes first-order data of $\mathcal {G}$ in directions normal to $\mathcal {L}$ (see also [Reference Crainic and StruchinerCS13]). Given $x\in \mathcal {L}$, so that $\mathcal {L}=\mathcal {L}_x$, this restricts to a representation

(10)\begin{equation} \mathcal{N}_x\in {\rm Rep}(\mathcal{G}_x) \end{equation}

of the isotropy group $\mathcal {G}_x$ on the fiber $\mathcal {N}_x$ of $\mathcal {N}_\mathcal {L}$ over $x$, which we refer to as the normal representation of $\mathcal {G}$ at $x$. Without loss of information, one can restrict attention to the normal representation at a point, which will often be more convenient for our purposes. This is because the transitive Lie groupoid $\mathcal {G}_\mathcal {L}$ is canonically isomorphic to the gauge groupoid of the principal bundle (8), and the normal bundle $\mathcal {N}_\mathcal {L}$ is canonically isomorphic to the vector bundle associated to the principal bundle (8) and the representation (10).

For a symplectic groupoid the basic facts stated below hold, which follow from multiplicativity of the symplectic form on the groupoid (see, e.g., [Reference Bursztyn, Crainic, Weinstein and ZhuBCWZ04] for background on multiplicative $2$-forms).

1.2.2 The orbits, leaves, and normal representations of Hamiltonian actions

Next, we will study the leaves and the normal representations for the action groupoid of a Hamiltonian action. Let $(\mathcal {G},\Omega )\rightrightarrows M$ be a symplectic groupoid and suppose that we are given a left Hamiltonian $(\mathcal {G},\Omega )$-action along $J:(S,\omega )\to M$. Let $p\in S$, $x:=J(p)\in M$, let $(\mathcal {L},\omega _\mathcal {L})$ be the symplectic leaf of $(\mathcal {G},\Omega )$ through $x$ (as in Proposition 1.10) and let $\mathcal {G}_x$ be the isotropy group of $\mathcal {G}$ at $x$. By the orbit of the action through $p$ we mean

\[ \mathcal{O}_p:=\{g\cdot p\mid g\in s^{-1}(x)\}\subset S, \]

and by the isotropy group of the $\mathcal {G}$-action at $p$ we mean the closed subgroup

\[ \mathcal{G}_p:=\{g\in \mathcal{G}_x\mid g\cdot p=p\}\subset \mathcal{G}_x. \]

Note that these coincide with the leaf and the isotropy group at $p$ of the action groupoid. We let

(11)\begin{equation} \mathcal{N}_p\in {\rm Rep}(\mathcal{G}_p) \end{equation}

denote the normal representation of the action groupoid at $p$. There are various relationships between the orbits, leaves and the normal representations at $p$ and $x$. To state these, consider the symplectic normal space to the orbit $\mathcal {O}$ at $p$:

(12)\begin{equation} \mathcal{S}\mathcal{N}_p:=\frac{T_p\mathcal{O}^\omega}{T_p\mathcal{O}\cap T_p\mathcal{O}^\omega}, \end{equation}

where we denote the symplectic orthogonal of the tangent space $T_p\mathcal {O}$ to the orbit through $p$ as

(13)\begin{equation} T_p\mathcal{O}^\omega:=\{v\in T_pS\mid\omega(v,w)=0,\ \forall w\in T_p\mathcal{O}\}. \end{equation}

Further, consider the annihilator of $\mathfrak {g}_p$ in $\mathfrak {g}_x$:

(14)\begin{equation} \mathfrak{g}_p^0\subset \mathfrak{g}_x^*. \end{equation}

Proof. That $J$ maps $\mathcal {O}$ submersively onto a leaf $\mathcal {L}$ follows from the axioms of a Lie groupoid action. The equality (15) is readily derived from (5). Part (c) is immediate from Proposition 1.10(b). To prove part (b) and provide some further insight into part (c), observe that $J$ induces a $\mathcal {G}_p$-equivariant map:

\[ \underline{{\rm d} J}_p:\mathcal{N}_p\to \mathcal{N}_x. \]

Therefore, we have two short exact sequences of $\mathcal {G}_p$-representations:

(18)\begin{align} & 0\to {\rm Ker}(\underline{{\rm d} J}_p)\to \mathcal{N}_p\to {\rm Im}(\underline{{\rm d} J}_p)\to 0 , \end{align}

(19)\begin{align} & 0\to {\rm Im}(\underline{{\rm d} J}_p)\to \mathcal{N}_x\to \text{CoKer}(\underline{{\rm d} J}_p)\to 0. \end{align}

Using the following proposition, the short exact sequence (19) translates into the short exact sequence (17), whereas (18) translates into (16). In particular, this proves part (b).

This is readily derived from the momentum map condition (6).

1.2.3 The symplectic normal representation

Note that the symplectic form $\omega$ on $S$ descends to a linear symplectic form $\omega _p$ on the symplectic normal space (12).

Proof. We ought to show that $\omega _p$ is $\mathcal {G}_p$-invariant. Note that, for any $v\in {\rm Ker}({\rm d} J_p)$ and $g\in \mathcal {G}_p$,

\[ g\cdot[v]=[{\rm d} m_{(g,p)}(0,v)]. \]

Thus, using Proposition 1.12(a) we find that for all $v,w\in T_p\mathcal {O}^\omega$ and $g\in \mathcal {G}_p$,

\[ \omega_p(g\cdot[v],g\cdot[w])=(m^*\omega)_{(g,p)}((0,v),(0,w))=\omega_p([v],[w]), \]

where in the last step we applied (5).

Definition 1.14 Given a Hamiltonian action as above, we call

(20)\begin{equation} (\mathcal{S}\mathcal{N}_p,\omega_p)\in {\rm SympRep}(\mathcal{G}_p) \end{equation}

its symplectic normal representation at $p$.

Given any symplectic representation $(V,\omega _V\!)$ of a Lie group $H$, the $H$-action is Hamiltonian with quadratic momentum map

(21)\begin{equation} J_V:(V,\omega_V\!)\to \mathfrak{h}^*,\quad \langle J_V(v),\xi \rangle=\tfrac{1}{2}\omega_V(\xi\cdot v,v). \end{equation}

As we will now show, given a Hamiltonian $(\mathcal {G},\Omega )$-action along $J:(S,\omega )\to M$, the quadratic momentum map,

(22)\begin{equation} J_{\mathcal{S}\mathcal{N}_p}:(\mathcal{S}\mathcal{N}_p,\omega_p)\to \mathfrak{g}_p^*, \end{equation}

of the symplectic normal representation at $p$ can be expressed in terms of the quadratic differential of $J$ at $p$. Recall from [Reference Arnold, Gusein-Zade and VarchenkoAGV12] that the quadratic differential of a map $F:S\to M$ at $p\in S$ is defined to be the quadratic map

\[ {\rm d}^2F_p:{\rm Ker}({\rm d} F_p)\to \text{CoKer}({\rm d} F_p),\quad {\rm d}^2F_p(v)= \bigg[\frac{1}{2}\frac{{\rm d}^2}{{\rm d}^2 t}\bigg|_{t=0}(\psi\circ F\circ \varphi^{-1})(tv)\bigg], \]

where $\varphi :(U,p)\to (T_pS,0)$ and $\psi :(V,x)\to (T_{x}M,0)$ are any two open embeddings, defined on open neighbourhoods of $p$ and $x:=F(p)$ such that $F(U)\subset V$, with the property that their differentials at $p$ and $x$ are the respective identity maps. Returning to the momentum map $J$, by Proposition 1.12 its quadratic differential becomes a map

(23)\begin{equation} {\rm d}^2J_p:T_p\mathcal{O}^\omega\to \mathfrak{g}_p^*. \end{equation}

Proposition 1.15 Let $J:(S,\omega )\to M$ be the momentum map of a Hamiltonian action and $p\in S$. Then the quadratic differential (23) is the composition of the quadratic momentum map (22) with the canonical projection $T_p\mathcal {O}^\omega \to \mathcal {S}\mathcal {N}_p$:

For the proof, we use an alternative description of the quadratic differential. Recall that, given a vector bundle $E\to S$ and a germ of sections $e\in \Gamma _p(E)$ vanishing at $p\in S$, the linearization of $e$ at $p$ is the linear map

\[ e^{\rm lin}_p:T_pS \to E_p,\quad e^{\rm lin}_p:={\rm pr}_{E_p}\circ ({\rm d} e)_p, \]

where we view the differential $({\rm d} e)_p$ of the map $e$ at $p$ as map into $E_p\oplus T_pS$, via the canonical identification of $(TE)_{(p,0)}$ with $E_p\oplus T_pS$. With this, one can define the intrinsic Hessian of $F$ at $p$ to be the symmetric bilinear map

\[ {\rm Hess}_p(F):{\rm Ker}({\rm d} F_p)\times {\rm Ker}({\rm d} F_p)\to \text{CoKer}({\rm d} F_p),\quad (X_p,Y_p)\mapsto\big[\tfrac{1}{2} ({\rm d} F(Y))^{\rm lin}_p(X_p)\big], \]

where $Y\in \mathcal {X}_p(S)$ is any germ of vector fields extending $Y_p$ and we see ${\rm d} F(Y)$ as a germ of sections of $F^*(TM)$. The quadratic differential is now given by the quadratic form

\[ {\rm d}^2F_p(v)={\rm Hess}_p(F)(v,v),\quad v\in {\rm Ker}({\rm d} F_p). \]

We will further use the following immediate, but useful, observation.

Lemma 1.16 Let $\Phi :E\to F$ be a map of vector bundles over the same manifold, covering the identity map. If $e\in \Gamma _p(E)$ is a germ of sections that vanishes at $p$, then so does $\Phi (e)\in \Gamma _p(F)$ and we have

\[ \Phi(e)^{\rm lin}_p=\Phi\circ e^{\rm lin}_p. \]

Proof of Proposition 1.15 Let $\alpha _x\in \mathfrak {g}_p \subset T_x^*M$ and $X_p\in {\rm Ker}({\rm d} J_p)=T_p\mathcal {O}^\omega$. We have to prove

\[ \langle J_{\mathcal{S}\mathcal{N}_p}([X_p]), \alpha_x\rangle=\langle \alpha_x, {\rm d}^2J_p(X_p)\rangle. \]

This will follow by linearizing both sides of (6). Let $\alpha \in \Omega ^1(M)$ and $X\in \mathcal {X}(S)$ be extensions of $\alpha _x$ and $X_p$, respectively. On one hand, we have

\begin{align*} \langle (\iota_{a(\alpha)}\omega)^{\rm lin}_p(X_p),X_p\rangle &= \omega_p(a(\alpha)^{\rm lin}_p(X_p),X_p)\\ &=2\langle J_{\mathcal{S}\mathcal{N}_p}([X_p]), \alpha_x\rangle. \end{align*}

Here we have first used that, given a $k$-form $\beta$ and a vector field $Y$ that vanishes at $p$, it holds that

\[ (\iota_Y\beta)^{\rm lin}_p(X_p)=\iota_{Y^{\rm lin}_p(X_p)}\beta_p, \]

as follows from Lemma 1.16. Furthermore, for the second step we have used that the Lie algebra representation $\mathfrak {g}_p\to \mathfrak {sp}(\mathcal {S}\mathcal {N}_p,\omega _p)$ induced by the symplectic normal representation is given by

\[ \alpha_x\cdot[X_p]=[a(\alpha)^{{\rm lin}}_p(X_p)]. \]

On the other hand, linearizing the right-hand side of (6) we find (as desired)

\begin{align*} \langle (J^*\alpha)^{\rm lin}_p(X_p),X_p\rangle&=(\alpha({\rm d} J(X))^{\rm lin}_p(X_p)\\ &=2\langle \alpha_x, {\rm d}^2J_p(X_p)\rangle. \end{align*}

Here we have first used that, given a vector field $Y$ and a $k$-form $\beta$ that vanishes at $p$, it holds that

\[ (\iota_Y\beta)^{\rm lin}_p(X_p)=\iota_{Y_p}(\beta^{\rm lin}_p(X_p)), \]

as follows from Lemma 1.16. Furthermore, for the second step we have again used Lemma 1.16.

1.2.4 Neighbourhood equivalence and rigidity

We now turn to the notion of neighbourhood equivalence, used in the statement of Theorem 1.1. In view of Proposition 1.10, the restriction of a symplectic groupoid $(\mathcal {G},\Omega )$ to a leaf $\mathcal {L}$ gives rise to the data of:

• • a symplectic manifold $(\mathcal {L},\omega _\mathcal {L})$;

• • a transitive Lie groupoid $\mathcal {G}_\mathcal {L}\rightrightarrows \mathcal {L}$ equipped with a closed multiplicative $2$-form $\Omega _\mathcal {L}$;

subject to the relation

(24)\begin{equation} \Omega_\mathcal{L}=t_{\mathcal{G}_\mathcal{L}}^*\omega_\mathcal{L}-s_{\mathcal{G}_\mathcal{L}}^*\omega_\mathcal{L}. \end{equation}

Further, using Proposition 1.11(a), we observe that the restriction of a Hamiltonian $(\mathcal {G},\Omega )$-action along $J:(S,\omega )\to M$ to an orbit $\mathcal {O}$ (with corresponding leaf $\mathcal {L}=J(\mathcal {O})$) encodes the data of:

• • a zeroth-order symplectic groupoid data $(\mathcal {G}_\mathcal {L},\Omega _\mathcal {L})\rightrightarrows (\mathcal {L},\omega _\mathcal {L})$;

• • a pre-symplectic manifold $(\mathcal {O},\omega _\mathcal {O})$;

• • a transitive Lie groupoid action of $\mathcal {G}_\mathcal {L}$ along a map $J_\mathcal {O}:\mathcal {O}\to \mathcal {L}$;

subject to the relations

(25)\begin{equation} ({\rm pr}_{\mathcal{G}_\mathcal{L}})^*\Omega_\mathcal{L}=(m_\mathcal{O})^*\omega_\mathcal{O}-({\rm pr}_\mathcal{O})^*\omega_\mathcal{O} \quad \text{and}\quad \omega_\mathcal{O}=(J_\mathcal{O})^*\omega_\mathcal{L}. \end{equation}

Here we use

\[ m_\mathcal{O},{\rm pr}_\mathcal{O}:\mathcal{G}_\mathcal{L}\ltimes \mathcal{O}\to \mathcal{O},\quad {\rm pr}_{\mathcal{G}_\mathcal{L}}:\mathcal{G}_\mathcal{L}\ltimes \mathcal{O}\to \mathcal{G}_\mathcal{L}, \]

to denote the map defining the action and the projections from the action groupoid to $\mathcal {O}$ and $\mathcal {G}_\mathcal {L}$.

Next, we define realizations of such zeroth-order data and neighbourhood equivalences thereof.

Definition 1.19 By a realization of a given zeroth-order symplectic groupoid data

we mean an embedding of Lie groupoids $i:\mathcal {G}_\mathcal {L}\hookrightarrow \mathcal {G}$ with the property that $\Omega$ pulls back to $\Omega _\mathcal {L}$ and that $\mathcal {G}_\mathcal {L}$ embeds as the restriction of $\mathcal {G}$ to a leaf. Of course, $(\mathcal {L},\omega _\mathcal {L})$ then automatically embeds as a symplectic leaf of $(\mathcal {G},\Omega )$. We call two realizations $i_1$ and $i_2$ of the same zeroth-order symplectic groupoid data neighbourhood-equivalent if there are opens $V_1$ and $V_2$ around $\mathcal {L}$ in $M_1$ and $M_2$ respectively, together with an isomorphism of symplectic groupoids:

that intertwines $i_1$ with $i_2$.

Definition 1.20 By a realization of a given zeroth-order Hamiltonian data

we mean a pair $(i,j)$ consisting of:

• • a realization $i$ of the zeroth-order symplectic groupoid data $(\mathcal {G}_\mathcal {L},\Omega _\mathcal {L})\rightrightarrows (\mathcal {L},\omega _\mathcal {L})$;

• • an embedding $j:\mathcal {O}\hookrightarrow S$ that pulls back $\omega$ to $\omega _\mathcal {O}$ and is compatible with $i$, in the sense that $i$ and $j$ together intertwine $J_\mathcal {O}$ with $J$, and the actions along these maps.

We call two realizations $(i_1,j_1)$ and $(i_2,j_2)$ of the same zeroth-order Hamiltonian data neighbourhood-equivalent if there are opens $V_1$ and $V_2$ around $\mathcal {L}$ in $M_1$ and $M_2$, respectively, a $\mathcal {G}_1\vert _{V_1}$-invariant open $U_1$ and a $\mathcal {G}_2\vert _{V_2}$-invariant open $U_2$ around $\mathcal {O}$ in $J_1^{-1}(V_1)$, respectively $J_2^{-1}(V_2)$, together with:

• • an isomorphism $(\mathcal {G}_1,\Omega _1)\vert _{V_1}\cong (\mathcal {G}_2,\Omega _2)\vert _{V_2}$ that intertwines $i_1$ with $i_2$;

• • a symplectomorphism $(U_1,\omega _1)\cong (U_2,\omega _2)$ that intertwines $j_1$ with $j_2$ and is compatible with the above isomorphism of symplectic groupoids, in the sense that together these intertwine $J_1:U_1\to V_1$ with $J_2:U_2\to V_2$, and the actions along these maps.

In other words, we have an isomorphism of Hamiltonian actions

that intertwines the embeddings of zeroth-order data. Usually the embeddings are clear from the context and we simply call the two Hamiltonian actions neighbourhood-equivalent around $\mathcal {O}$.

We can now state the rigidity result mentioned in the introduction to this section.

In the coming section, we give an explicit construction to show the following.

Given a Hamiltonian action, we call the realization constructed from the zeroth-order Hamiltonian data obtained by restriction to $\mathcal {O}$ and from the symplectic normal representation at $p$: the local model of the Hamiltonian action around $\mathcal {O}$ (we disregard the choice of $p\in \mathcal {O}$, as different choices result in isomorphic local models). Applying Theorem 1.21 to the given Hamiltonian action on one hand and, on the other hand, to its local model around $\mathcal {O}$, Theorem 1.1 follows. Hence, after the construction of this local model, it remains for us to prove Theorem 1.21.

1.3.1 Reorganization of the zeroth-order Hamiltonian data

Before constructing the local model, we rearrange the zeroth-order data (defined in the previous subsection) into a simpler form. First, due to the relations (24) and (25), the triple of $2$-forms $\Omega _\mathcal {L}$, $\omega _\mathcal {L}$ and $\omega _\mathcal {O}$ can be fully reconstructed from the single $2$-form $\omega _\mathcal {L}$. Therefore, a collection of zeroth-order Hamiltonian data can equivalently be defined as the data of:

• • a symplectic manifold $(\mathcal {L},\omega _\mathcal {L})$;

• • a transitive Lie groupoid $\mathcal {G}_\mathcal {L}\rightrightarrows \mathcal {L}$;

• • a transitive Lie groupoid action of $\mathcal {G}_\mathcal {L}$ along a map $J_\mathcal {O}:\mathcal {O}\to \mathcal {L}$.

After the choice of a point $p\in \mathcal {O}$, this can be simplified further to a collection consisting of:

• • a symplectic manifold $(\mathcal {L},\omega _\mathcal {L})$;

• • a Lie group $G$ (corresponding to $\mathcal {G}_x$);

• • a (right) principal $G$-bundle $P\to \mathcal {L}$ (corresponding to $t:s^{-1}(x)\to \mathcal {L}$);

• • a closed subgroup $H$ of $G$ (corresponding to $\mathcal {G}_p$).

To see this, fix a point $p\in \mathcal {O}$ and let $x=J_\mathcal {O}(p)\in \mathcal {L}$. Since $\mathcal {G}_\mathcal {L}$ is transitive, the choice of $x\in \mathcal {L}$ induces an isomorphism between $\mathcal {G}_\mathcal {L}$ and the gauge groupoid

(26)\begin{equation} s^{-1}(x)\times_{\mathcal{G}_x}s^{-1}(x) \rightrightarrows \mathcal{L}, \end{equation}

of the principal $\mathcal {G}_x$-bundle $t:s^{-1}(x)\to \mathcal {L}$. In particular, $\mathcal {G}_\mathcal {L}$ is entirely encoded by this principal bundle. Furthermore, due to transitivity the $\mathcal {G}_\mathcal {L}$-action along $J_\mathcal {O}$ is entirely determined by this principal bundle and the subgroup $\mathcal {G}_p$ of $\mathcal {G}_x$. Indeed, the map $J_\mathcal {O}$ can be recovered from this, for we have a commutative square

where the left vertical map is defined by acting on $p$ and the upper horizontal map is the canonical one. Moreover, the action can be recovered as the action of the groupoid (26) along the upper horizontal map, given by $[p,q]\cdot [q]=[p]$.

1.3.2 Construction of the local model for the symplectic groupoid

The construction presented here is well known. For other (more Poisson geometric) constructions of this local model, see [Reference Crainic and MarcutCM12, Reference MarcutMar13]. The local model for the symplectic groupoid is built out of the zeroth-order symplectic groupoid data, encoded as above by:

• • a symplectic manifold $(\mathcal {L},\omega _\mathcal {L})$;

• • a Lie group $G$;

• • a (right) principal $G$-bundle $P\to \mathcal {L}$.

To construct the local model, we make an auxiliary choice of a connection $1$-form $\theta \in \Omega ^1(P;\mathfrak {g})$ and define

(27)\begin{equation} \hat{\theta}\in \Omega^1(P\times \mathfrak{g}^*),\quad \hat{\theta}_{(q,\alpha)}=\langle \alpha, \theta_q\rangle. \end{equation}

Then, we use the symplectic structure $\omega _\mathcal {L}$ on $\mathcal {L}$ to define

(28)\begin{equation} \omega_\theta=({\rm pr}_\mathcal{L})^*\omega_\mathcal{L}-{\rm d}\hat{\theta}\in \Omega^2(P\times \mathfrak{g}^*), \end{equation}

where by ${\rm pr}_\mathcal {L}$ we denote the composition $P\times \mathfrak {g}^*\xrightarrow {{\rm pr}_1} P\to \mathcal {L}$. The $2$-form $\omega _\theta$ is closed, non-degenerate at all points of $P\times \{0\}$ and $(P\times \mathfrak {g}^*,\omega _\theta )\to \mathfrak {g}^*$ is a (right) pre-symplectic Hamiltonian $G$-space. Therefore, the open subset $\Sigma _\theta \subset P\times \mathfrak {g}^*$ on which $\omega _\theta$ is non-degenerate is a $G$-invariant neighbourhood of $P\times \{0\}$. Since the action is free and proper, the symplectic form $\omega _\theta$ descends to a Poisson structure $\pi _\theta$ on the open neighbourhood $M_\theta$ of the zero-section $\mathcal {L}$, defined as

\begin{align*} M_\theta:=\Sigma_\theta/G\subset P\times_G\mathfrak{g}^*. \end{align*}

This is the base of the local model. For the construction of the integrating symplectic groupoid, notice first that the pair groupoid

(29)\begin{equation} (\Sigma_\theta\times \Sigma_\theta,\omega_\theta\oplus -\omega_\theta) \end{equation}

is a symplectic groupoid and, furthermore, it is a (right) free and proper Hamiltonian $G$-space (being a product of two). Therefore, the symplectic form $\omega _\theta \oplus -\omega _\theta$ descends to the symplectic reduced space at $0\in \mathfrak {g}^*$:

(30) \begin{align} (\mathcal{G}_\theta,\Omega_\theta):=((\Sigma_\theta\times\Sigma_\theta)// G,\Omega_{\rm red}). \end{align}

The pair groupoid structure on $\Sigma _\theta \times \Sigma _\theta$ descends to a Lie groupoid structure on (30), making it a symplectic groupoid integrating $(M_\theta,\pi _\theta )$. This is the symplectic groupoid in the local model. It is canonically a realization of the given zeroth-order symplectic groupoid data: the gauge groupoid of the principal $G$-bundle $P\to \mathcal {L}$ (corresponding to (26)) embeds into (30) via the zero-section.

1.3.3 Construction of the local model for Hamiltonian actions

The construction below generalizes that in [Reference MarleMar85, Reference Guillemin and SternbergGS84]. The local model is built out of a zeroth-order Hamiltonian data and a symplectic representation of an isotropy group of the action, encoded as in § 1.3.1 by:

• • a symplectic manifold $(\mathcal {L},\omega _\mathcal {L})$;

• • a Lie group $G$;

• • a (right) principal $G$-bundle $P\to \mathcal {L}$;

• • a closed subgroup $H$ of $G$;

• • a symplectic $H$-representation $(V,\omega _V\!)$.

Choose an auxiliary connection $1$-form $\theta \in \Omega ^1(P;\mathfrak {g})$ and define $\omega _\theta$, $\Sigma _\theta$ and $M_\theta$ as in the construction of the local model for symplectic groupoids. To construct a Hamiltonian action of the symplectic groupoid (30), consider the product of the Hamiltonian $H$-spaces:

\[ {\rm pr}_{\mathfrak{h}^*}:(\Sigma_\theta,\omega_\theta)\xrightarrow{{\rm pr}_{\mathfrak{g}^*}}\mathfrak{g}^*\to \mathfrak{h}^*\quad \mbox{and} \quad J_V:(V,\omega_V\!)\to \mathfrak{h}^*, \]

where $J_V$ is as in (21). This is another (right) Hamiltonian $H$-space:

\[ J_H:(\Sigma_\theta\times V, {\omega}_\theta\oplus\omega_V\!)\to \mathfrak{h}^*,\quad (q,\alpha,v)\mapsto \alpha\vert_\mathfrak{h}-J_V(v), \]

where the action is the diagonal one, which is free and proper. The symplectic manifold in the local model is the reduced space at $0\in \mathfrak {h}^*$:

(31)\begin{equation} (S_\theta,\omega_{S_\theta}):=((\Sigma_\theta\times V) // H,\omega_{{\rm red}}). \end{equation}

To equip this with a Hamiltonian action of (30), observe that, on the other hand, the symplectic pair groupoid (29) acts along

\[ \text{pr}_{\Sigma_\theta}:(\Sigma_\theta\times V, {\omega}_\theta\oplus\omega_V\!)\to \Sigma_\theta \]

in a Hamiltonian fashion as $(\sigma,\tau )\cdot (\tau,v)=(\sigma,v)$ for $\sigma,\tau \in \Sigma _\theta$ and $v\in V$. This descends to a Hamiltonian action of (30) that fits into a diagram of commuting Hamiltonian actions:

with the property that the momentum map of each one is invariant under the action of the other. It therefore follows that the left-hand action descends to a Hamiltonian action along the map

(32)\begin{equation} J_\theta:(S_\theta,\omega_{S_\theta})\to M_\theta, \quad [\sigma,v]\mapsto [\sigma]. \end{equation}

This is the Hamiltonian action in the local model. It is canonically a realization of the given zeroth-order Hamiltonian data: as in the previous subsection the gauge groupoid of the principal $G$-bundle $P\to \mathcal {L}$ embeds into (30) via the zero-section and similarly $P/H$ embeds into (31). This completes the construction of the local model. Finally, given the starting data in Proposition 1.22, one readily verifies that the symplectic normal representation at $p$ of the resulting Hamiltonian action of (30) along (32) is isomorphic to $(V,\omega _V\!)$ as symplectic $\mathcal {G}_p$-representation. Thus, this also completes the proof of Proposition 1.22.

Remark 1.23 Under the assumption that the short exact sequence

(33)\begin{equation} 0\to \mathfrak{h}^0\to \mathfrak{g}^*\to \mathfrak{h}^*\to 0 \end{equation}

splits $H$-equivariantly (which holds if $H$ is compact), the local model can be put in the more familiar form of a vector bundle over $\mathcal {O}$. Indeed, let $\mathfrak {p}:\mathfrak {h}^*\to \mathfrak {g}^*$ be such a splitting. Then we have an open embedding

(34)\begin{equation} S_\theta\to P\times_H(\mathfrak{h}^0\oplus V),\quad [p,\alpha,v]\mapsto [p,\alpha-\mathfrak{p}(J_V(v)),v] \end{equation}

onto an open neighbourhood of the zero-section, which identifies the momentum map (32) with the restriction to this open neighbourhood of the map

(35)\begin{equation} P\times_H(\mathfrak{h}^0\oplus V)\to P\times_G\mathfrak{g}^*,\quad [p,\alpha,v]\mapsto [p,\alpha+\mathfrak{p}(J_V(v))]. \end{equation}

To identify the action accordingly, observe that, as Lie groupoid, (30) embeds canonically onto an open subgroupoid of

(36)\begin{equation} (P\times P)\times_{G}\mathfrak{g}^*\rightrightarrows P\times_G\mathfrak{g}^*, \end{equation}

which inherits its Lie groupoid structure from the submersion groupoid of ${\rm pr}_{\mathfrak {g}^*}:P\times \mathfrak {g}^*\to \mathfrak {g}^*$, being a quotient of it. This identifies the action of (30) along (35) with (a restriction of) the action of (36) along (35), given by

\[ [p_1,p_2,\alpha+\mathfrak{p}(J_V(v))]\cdot [p_2,\alpha,v]=[p_1,\alpha,v],\quad p_1,p_2\in P,\quad \alpha\in \mathfrak{h}^0,\quad v\in V. \]

1.3.4 Relation to the MGS model

Let $G$ be a Lie group and consider a Hamiltonian $G$-space $J:(S,\omega )\to \mathfrak {g}^*$. As remarked in Example 1.5, this is the same as a Hamiltonian action of the cotangent groupoid $(G\ltimes \mathfrak {g}^*,-{\rm d} \lambda _{{\rm can}})\rightrightarrows \mathfrak {g}^*$ along $J$. Let $p\in S$, $\alpha =J(p)$ and suppose that $G\ltimes \mathfrak {g}^*$ is proper at $\alpha$ (in the sense of Definition 1.7). In this case, our local model around the orbit $\mathcal {O}$ through $p$ is equivalent to the local model in the MGS normal form theorem for Hamiltonian $G$-spaces (recalled below). To see this, first note that, since the isotropy group $G_\alpha$ is compact, the short exact sequence of $G_\alpha$-representations

(37)\begin{equation} 0\to \mathfrak{g}_\alpha^0\to \mathfrak{g}^*\to \mathfrak{g}_\alpha^*\to 0 \end{equation}

is split. Let $\sigma :\mathfrak {g}_\alpha ^*\to \mathfrak {g}^*$ be a $G_\alpha$-equivariant splitting of (37) and consider the connection one-form $\theta \in \Omega ^1(G;\mathfrak {g}_\alpha )$ on $G$ (viewed as right principal $G_\alpha$-bundle) obtained by composing the left-invariant Maurer–Cartan form on $G$ with $\sigma ^*:\mathfrak {g}\to \mathfrak {g}_\alpha$. The leaf $\mathcal {L}$ through $\alpha$ is a coadjoint orbit and $\omega _{\mathcal {L}}$ is the Kirillov–Kostant–Souriau symplectic form, which is invariant under the coadjoint action. Therefore, the $2$-form $\omega _\theta \in \Omega ^2(G\times \mathfrak {g}^*_\alpha )$, defined as in (28), is not only invariant under the right diagonal action of $G_\alpha$, but it also invariant under the left action of $G$ by left translation on the first factor. This implies that the open $\Sigma _\theta$ on which $\omega _\theta$ is non-degenerate is of the form $G\times W$ for a $G_\alpha$-invariant open $W$ around the origin in $\mathfrak {g}_\alpha ^*$. The local model for the cotangent groupoid around $\mathcal {L}$ becomes

\begin{align*} (G\ltimes (G\times_{G_\alpha} W), \Omega_\theta)\rightrightarrows G\times_{G_\alpha}W, \end{align*}

the groupoid associated to the action of $G$ by left translation on the first factor. To compare this to the cotangent groupoid itself, consider the $G$-equivariant map

\[ \varphi:G\times_{G_\alpha}W\to \mathfrak{g}^*,\quad [g,\beta]\mapsto g\cdot(\alpha+\sigma(\beta)). \]

Since $G\ltimes \mathfrak {g}^*$ is proper at $\alpha$, we can shrink $W$ so that $\varphi$ becomes an embedding onto a $G$-invariant open neighbourhood of $\mathcal {L}$. Then $\varphi$ lifts canonically to an isomorphism of symplectic groupoids

(38)\begin{equation} (G\ltimes (G\times_{G_\alpha}W), \Omega_\theta)\xrightarrow{\sim} (G\ltimes \mathfrak{g}^*,-{\rm d}\lambda_{{\rm can}})\vert_{\varphi(G\times_{G_\alpha}W)}, \end{equation}

and this is a neighbourhood equivalence around $G\ltimes \mathcal {L}$ (with respect to the canonical embeddings). Our local model for $(S,\omega )$ around $\mathcal {O}$ is the same as that in the MGS local model, and via (38) the Hamiltonian action in our local model is identified with the Hamiltonian $G$-space in the MGS local model. In particular the momentum map (35) is identified with

\[ J_{{\rm MGS}}:G\times_{G_p}(\mathfrak{g}_p^0\oplus \mathcal{S}\mathcal{N}_p)\to \mathfrak{g}^*,\quad [g,\beta,v]\mapsto g\cdot(\alpha+\sigma(\beta+\mathfrak{p}(J_{\mathcal{S}\mathcal{N}_p}(v)))). \]

1.4.1 Morita equivalence of groupoids

To prove Theorem 1.21 (and, hence, Theorem 1.1), we will reduce to the case where $\mathcal {O}\subset J_X^{-1}(0)$ is an orbit of a Hamiltonian $G$-space $J_X:(X,\omega _X\!)\to \mathfrak {g}^*$ (with $G$ a compact Lie group), to which we can apply the MGS theorem. The idea of such a reduction is by no means new: in fact, it has already appeared in the work of Guillemin and Sternberg. To do so, we use the fact that Morita equivalent symplectic groupoids have equivalent categories of modules. In preparation for this, we will now first recall the definition, some useful properties and examples of Morita equivalence.

Definition 1.25 Let $\mathcal {G}_1\rightrightarrows M_1$ and $\mathcal {G}_2\rightrightarrows M_2$ be Lie groupoids. A Morita equivalence from $\mathcal {G}_1$ to $\mathcal {G}_2$ is a principal $(\mathcal {G}_1,\mathcal {G}_2)$-bi-bundle $(P,\alpha _1,\alpha _2)$. This consists of:

• • a manifold $P$ with two surjective submersions $\alpha _i:P\to M_i$;

• • a left action of $\mathcal {G}_1$ along $\alpha _1$ that makes $\alpha _2$ into a principal $\mathcal {G}_1$-bundle;

• • a right action of $\mathcal {G}_2$ along $\alpha _2$ that makes $\alpha _1$ into a principal $\mathcal {G}_2$-bundle.

Furthermore, the two actions are required to commute. We depict this as follows.

For every leaf $\mathcal {L}_1\subset M_1$, there is a unique leaf $\mathcal {L}_2\subset M_2$ such that $\alpha _1^{-1}(\mathcal {L}_1)=\alpha _2^{-1}(\mathcal {L}_2)$; such leaves $\mathcal {L}_1$ and $\mathcal {L}_2$ are called $P$-related. When $(\mathcal {G}_1,\Omega _1)$ and $(\mathcal {G}_2,\Omega _2)$ are symplectic groupoids, then a symplectic Morita equivalence from $(\mathcal {G}_1,\Omega _1)$ to $(\mathcal {G}_2,\Omega _2)$ is a Morita equivalence with the extra requirement that $(P,\omega _P)$ is a symplectic manifold and both actions are Hamiltonian.

Morita equivalence is an equivalence relation that, heuristically speaking, captures the geometry transverse to the leaves. The simplest motivation for this principle is the following basic result.

Example 1.30 The following example plays a crucial role in our proof of Theorem 1.21. Consider the set-up of § 1.3.2. There is a canonical symplectic Morita equivalence

between (30) and the restriction of the cotangent groupoid to the $G$-invariant open $W_\theta :={\rm pr}_{\mathfrak {g}^*}(\Sigma _\theta )$ around the origin in $\mathfrak {g}^*$. This relates the central leaf $\mathcal {L}$ in $M_\theta$ to the origin in $\mathfrak {g}^*$.

1.4.2 Equivalence between categories of modules

Next, we recall how a Morita equivalence induces an equivalence between the categories of modules. Given a Lie groupoid $\mathcal {G}\rightrightarrows M$, by a $\mathcal {G}$-module we simply mean a smooth map $J:S\to M$ equipped with a left action of $\mathcal {G}$. A morphism from a $\mathcal {G}$-module $J_1:S_1\to M$ to $J_2:S_2\to M$ is a smooth map $\varphi :S_1\to S_2$ that intertwines $J_1$ and $J_2$ and is $\mathcal {G}$-equivariant. This defines a category $\textsf {Mod}(\mathcal {G})$.

We now recall the following.

Proof. To any $\mathcal {G}_1$-module $J:S\to M_1$ we can associate a $\mathcal {G}_2$-module, as follows. The Lie groupoid $\mathcal {G}_1$ acts diagonally on the manifold $P\times _{M_1} S$ along the map $\alpha _1\circ {\rm pr}_1$, in a free and proper way. Hence, the quotient

\[ P\ast_{\mathcal{G}_1} S:=\frac{(P\times_{M_1} S)}{\mathcal{G}_1} \]

is smooth. Moreover, since the actions of $\mathcal {G}_1$ and $\mathcal {G}_2$ commute and $\alpha _1$ is $\mathcal {G}_2$-invariant, we have a left action of $\mathcal {G}_2$ along

(42)\begin{equation} P_*(J):P\ast_{\mathcal{G}_1} S\to M_2, \quad [p_P,p_S]\mapsto \alpha_2(p_P), \end{equation}

given by

\[ g\cdot [p_P,p_S]=[p_P\cdot g^{-1},p_S]. \]

We call this the $\mathcal {G}_2$-module associated to the $\mathcal {G}_1$-module $J$. For any morphism of $\mathcal {G}_1$-modules there is a canonical morphism between the associated $\mathcal {G}_2$-modules. Thus, this defines a functor

(43)\begin{equation} \begin{aligned} \textsf{Mod}(\mathcal{G}_1) & \to\textsf{Mod}(\mathcal{G}_2) \\ (J:S\to M_1) & \mapsto (P_*(J):P\ast_{\mathcal{G}_1}S\to M_2). \end{aligned} \end{equation}

An analogous construction from right to left gives an inverse to this functor.

Next, we recall the analogue for symplectic groupoids. Given a symplectic groupoid $(\mathcal {G},\Omega )\rightrightarrows M$, by a Hamiltonian $(\mathcal {G},\Omega )$-space (called symplectic left $(\mathcal {G},\Omega )$-module in [Reference XuXu91]) we mean a smooth map $J:(S,\omega )\to M$ equipped with a left Hamiltonian $(\mathcal {G},\Omega )$-action. A morphism $\varphi$ from $J_1:(S_1,\omega _1)\to M$ to $J_2:(S_2,\omega _2)\to M$ is a morphism of $\mathcal {G}$-modules satisfying $\varphi ^*\omega _2=\omega _1$. This defines a category $\textsf {Ham}(\mathcal {G},\Omega )$.

Proof. Let $(P,\omega _P,\alpha _1,\alpha _2)$ be a symplectic Morita equivalence between symplectic groupoids $(\mathcal {G}_1,\Omega _1)$ and $(\mathcal {G}_2,\Omega _2)$ and let $J:(S,\omega _S)\to M_1$ be a Hamiltonian $(\mathcal {G}_1,\Omega _1)$-space. The symplectic form $(-\omega _P)\oplus \omega _S$ descends to a symplectic form $\omega _{PS}$ on $P\ast _{\mathcal {G}_1}S$ and the $(\mathcal {G}_2,\Omega _2)$-action along the associated module $P_*(J)$, as in (42), becomes Hamiltonian. As before, this extends to a functor

(44)\begin{equation} \begin{aligned} \textsf{Ham}(\mathcal{G}_1,\Omega_1) & \to\textsf{Ham}(\mathcal{G}_2,\Omega_2) \\ (J:(S,\omega_S)\to M_1) & \mapsto (P_*(J):(P\ast_{\mathcal{G}_1}S,\omega_{PS})\to M_2) \end{aligned} \end{equation}

and an analogous construction from right to left gives an inverse functor.

1.4.3 Proof of rigidity

The proof of Theorem 1.21 hinges on the following two known results. The first is a rigidity theorem for symplectic groupoids.

The second result that we will need is the following rigidity theorem for Hamiltonian $G$-spaces.

Theorem 1.39 [Reference MarleMar83, Reference Guillemin and SternbergGS84]

Let $G$ be a compact Lie group and let $J_1:(S_1,\omega _1)\to \mathfrak {g}^*$ and $J_2:(S_2,\omega _2)\to \mathfrak {g}^*$ be Hamiltonian $G$-spaces. Suppose that $p_1\in J_1^{-1}(0)$ and $p_2\in J_2^{-1}(0)$ are such that $G_{p_1}=G_{p_2}$. Then there are $G$-invariant neighbourhoods $U_1$ of $p_1$ and $U_2$ of $p_2$, together with an isomorphism of Hamiltonian $G$-spaces that sends $p_1$ to $p_2$:

if and only if there is an equivariant symplectic linear isomorphism

\[ (\mathcal{S}\mathcal{N}_{p_1},\omega_{p_1})\cong (\mathcal{S}\mathcal{N}_{p_2},\omega_{p_2}). \]

The main step in proving Theorem 1.21 is to prove the following generalization of Theorem 1.39.

Theorem 1.40 Let $(\mathcal {G},\Omega )\rightrightarrows M$ be a symplectic groupoid that is proper at $x\in M$. Suppose that we are given two Hamiltonian $(\mathcal {G},\Omega )$-spaces $J_1:(S_1,\omega _1)\to M$ and $J_2:(S_2,\omega _2)\to M$. Let $p_1\in S_1$ and $p_2\in S_2$ be such that $J_1(p_1)=J_2(p_2)=x$ and $\mathcal {G}_{p_1}=\mathcal {G}_{p_2}$. Then there are $\mathcal {G}$-invariant open neighbourhoods $U_1$ of $p_1$ and $U_2$ of $p_2$, together with an isomorphism of Hamiltonian $(\mathcal {G},\Omega )$-spaces that sends $p_1$ to $p_2$:

if and only if there is an equivariant symplectic linear isomorphism

\[ (\mathcal{S}\mathcal{N}_{p_1},\omega_{p_1})\cong (\mathcal{S}\mathcal{N}_{p_2},\omega_{p_2}). \]

To prove this we further use the following lemma.

Lemma 1.41 Let $(P,\omega _P,\alpha _1,\alpha _2)$ be a symplectic Morita equivalence between $(\mathcal {G}_1,\Omega _1)$ and $(\mathcal {G}_2,\Omega _2)$. Further, let $J:(S,\omega _S)\to M$ be a Hamiltonian $(\mathcal {G}_1,\Omega _1)$-space, let $p_S\in S$ and fix a $p_P\in P$ such that $\alpha _1(p_P)=J(p_S)$. Then the isomorphism (40) restricts to an isomorphism

\[ \Phi_{p_P}:\mathcal{G}_{p_S}\xrightarrow{\sim} \mathcal{G}_{[p_P,p_S]}, \]

and there is a compatible symplectic linear isomorphism

\[ (\mathcal{S}\mathcal{N}_{p_S},(\omega_S)_{p_S})\cong(\mathcal{S}\mathcal{N}_{[p_P,p_S]},(\omega_{PS})_{[p_P,p_S]}) \]

between the symplectic normal representation at $p_S$ of the Hamiltonian $(\mathcal {G}_1,\Omega _1)$-space $J$ and the symplectic normal representation at $[p_P,p_S]$ of the associated Hamiltonian $(\mathcal {G}_2,\Omega _2)$-space $P_*(J)$ of Theorem 1.36.

Although this lemma can be verified directly, we postpone its proof to § 1.5.4, where we give a more conceptual explanation. With this at hand, we can prove the desired theorems.

Proof of Theorem 1.40 The forward implication is straightforward. Let us prove the backward implication. Throughout, let $G:=\mathcal {G}_x$ denote the isotropy group of $\mathcal {G}$ at $x$. To begin with observe that, since $\mathcal {G}$ is proper at $x$, there is an invariant open neighbourhood $V$ of the leaf $\mathcal {L}$ through $x$ and a $G$-invariant open neighbourhood $W$ of the origin in $\mathfrak {g}^*$, together with a symplectic Morita equivalence

that relates the leaf $\mathcal {L}$ to the origin in $\mathfrak {g}^*$. Indeed, this follows by first applying Theorem 1.37 to:

• • the zeroth-order data of $(\mathcal {G},\Omega )$ at $\mathcal {L}$;

• • the canonical realization $(\mathcal {G},\Omega )$;

• • the realization (30);

and combining the neighbourhood-equivalence of symplectic groupoids obtained thereby with Examples 1.28 and 1.30 to find a symplectic Morita equivalence as above in which the open around $\mathcal {L}$ is possibly not invariant; the invariance can then be ensured by composing with the Morita equivalence in Example 1.29. Since $V$ is $\mathcal {G}$-invariant, so are $J_1^{-1}(V)$ and $J_2^{-1}(V)$ and we can consider the Hamiltonian $(\mathcal {G},\Omega )$-spaces

(45)\begin{equation} {J_1^V}:(J_1^{-1}(V),\omega_1)\to M \quad \text{and}\quad {J_2^V}:(J_2^{-1}(V),\omega_2)\to M \end{equation}

obtained by restricting the given Hamiltonian $(\mathcal {G},\Omega )$-spaces $J_1$ and $J_2$. By Theorem 1.36, combined with Examples 1.34 and 1.35, the above Morita equivalence induces an equivalence of categories (with explicit inverse) between the category of Hamiltonian $(\mathcal {G},\Omega )$-spaces $J:(S,\omega )\to M$ with $J(S)\subset V$ and the category of Hamiltonian $G$-spaces $J:(S,\omega )\to \mathfrak {g}^*$ with $J(S)\subset W$. Consider the Hamiltonian $G$-spaces associated to (45),

\[ P_*(J_1^V):(P\ast_{(\mathcal{G}\vert_V\!)} J_1^{-1}(V),\omega_{PS_1})\to \mathfrak{g}^* \quad \mbox{and} \quad P_*(J_2^V):(P\ast_{(\mathcal{G}\vert_V\!)} J_2^{-1}(V),\omega_{PS_2})\to \mathfrak{g}^*, \]

and fix a $p\in P$ such that $\alpha _1(p)=x$. We will show that these Hamiltonian $G$-spaces satisfy the assumptions of Theorem 1.39 for the points $[p,p_1]$ and $[p,p_2]$. First of all, since the leaf $\mathcal {L}$ is $P$-related to the origin in $\mathfrak {g}^*$, it must be that $\alpha _2(p)=0$. Therefore, we find

\[ P_*(J_1^V)([p,p_1])=\alpha_2(p)=0 \quad \mbox{and} \quad P_*(J_2^V)([p,p_2])=\alpha_2(p)=0. \]

Second, Lemma 1.41 implies that $G_{[p,p_1]}=G_{[p,p_2]}$, as both coincide with the image of $\mathcal {G}_{p_1}=\mathcal {G}_{p_2}$ under $\Phi _p:\mathcal {G}_x\to G$. Third, by the same lemma, there are symplectic linear isomorphisms

\[ \psi_1: (\mathcal{S}\mathcal{N}_{p_1},(\omega_1)_{p_1})\xrightarrow{\sim} (\mathcal{S}\mathcal{N}_{[p,p_1]},(\omega_{PS_1})_{[p,p_1]})\enspace \mbox{and} \enspace \psi_2: (\mathcal{S}\mathcal{N}_{p_2},(\omega_2)_{p_2})\xrightarrow{\sim} (\mathcal{S}\mathcal{N}_{[p,p_2]},(\omega_{PS_2})_{[p,p_2]}), \]

that are both compatible with the isomorphism of Lie groups:

\[ \mathcal{G}_{p_1}\xrightarrow{\Phi_p} G_{[p,p_1]}=\mathcal{G}_{p_2}\xrightarrow{\Phi_p} G_{[p,p_2]}. \]

By assumption, there is an equivariant symplectic linear isomorphism

\[ \psi:(\mathcal{S}\mathcal{N}_{p_1},\omega_{p_1})\xrightarrow{\sim} (\mathcal{S}\mathcal{N}_{p_2},\omega_{p_2}). \]

All together, the composition

\[ \psi_2\circ \psi\circ \psi_1^{-1}: (\mathcal{S}\mathcal{N}_{[p,p_1]},(\omega_{PS_1})_{[p,p_1]}) \xrightarrow{\sim} (\mathcal{S}\mathcal{N}_{[p,p_2]},(\omega_{PS_2})_{[p,p_2]}) \]

becomes an equivariant symplectic linear isomorphism. Thus, the assumptions of Theorem 1.39 hold, which implies that there are $G$-invariant opens $U_{[p,p_1]}$ around $[p,p_1]$ and $U_{[p,p_2]}$ around $[p,p_2]$, together with an isomorphism of Hamiltonian $G$-spaces that sends $[p,p_1]$ to $[p,p_2]$:

One readily verifies that, by passing back through the above equivalence of categories via the explicit inverse functor, we obtain $\mathcal {G}$-invariant opens $U_1$ around $p_1$ and $U_2$ around $p_2$, together with an isomorphism of Hamiltonian $(\mathcal {G},\Omega )$-spaces from $J_1:(U_1,\omega _1)\to M$ to $J_2:(U_2,\omega _2)\to M$ that sends $p_1$ to $p_2$, as desired.

Proof of Theorem 1.21 As in the previous proof, the forward implication is straightforward. For the backward implication, let $(i_1,j_1)$ and $(i_2,j_2)$ be two realizations of the same zeroth-order Hamiltonian data (with notation as in Definition 1.20). Let $p\in \mathcal {O}$ and $x=J_\mathcal {O}(p)$ and suppose that their symplectic normal representations at $p$ are isomorphic as symplectic $\mathcal {G}_p$-representations. By Theorem 1.37 there are respective opens $V_1$ and $V_2$ around $\mathcal {L}$ in $M_1$ and $M_2$, together with an isomorphism

\[ \Phi:(\mathcal{G}_1,\Omega_1)\vert_{V_1}\xrightarrow{\sim}(\mathcal{G}_2,\Omega_2)\vert_{V_2} \]

that intertwines $i_1$ with $i_2$. Consider, on one hand, the Hamiltonian $(\mathcal {G}_1,\Omega _1)\vert _{V_1}$-space obtained from the given Hamiltonian $(\mathcal {G}_1,\Omega _1)$-space $J_1$ by restriction to $V_1$ and, on the other hand, the Hamiltonian $(\mathcal {G}_1,\Omega _1)\vert _{V_1}$-space $\Phi ^*(J_2)$ obtained from the given Hamiltonian $(\mathcal {G}_2,\Omega _2)$-space $J_2$ by restriction to $V_2$ and pullback along $\Phi$. These two Hamiltonian $(\mathcal {G}_1,\Omega _1)\vert _{V_1}$-spaces meet the assumptions of Theorem 1.40 at the points $j_1(p)$ and $j_2(p)$. Thus, there are $(\mathcal {G}_1\vert _{V_1})$-invariant opens $U_1\subset J_1^{-1}(V_1)$ and $U_2\subset J_2^{-1}(V_2)$, together with an isomorphism of Hamiltonian $(\mathcal {G}_1,\Omega _1)\vert _{V_1}$-spaces that sends $j_1(p)$ to $j_2(p)$:

As one readily verifies, the pair $(\Phi,\Psi )$ is the desired neighbourhood equivalence.

1.5.1 Hamiltonian Morita equivalence

In order to define a notion of Morita equivalence between Hamiltonian actions, we first consider a natural equivalence relation between Lie groupoid maps (respectively, groupoid maps of Hamiltonian type, defined below). In the next subsection we explain how this restricts to an equivalence relation between Lie groupoid actions (respectively, Hamiltonian actions).

Definition 1.42 Let $\mathcal {J}_1:\mathcal {H}_1\to \mathcal {G}_1$ and $\mathcal {J}_2:\mathcal {H}_2\to \mathcal {G}_2$ be maps of Lie groupoids. By a Morita equivalence from $\mathcal {J}_1$ to $\mathcal {J}_2$ we mean the data consisting of:

• • a Morita equivalence $(P,\alpha _1,\alpha _2)$ from $\mathcal {G}_1$ to $\mathcal {G}_2$;

• • a Morita equivalence $(Q,\beta _1,\beta _2)$ from $\mathcal {H}_1$ to $\mathcal {H}_2$;

• • a smooth map $j:Q\to P$ that intertwines $J_i\circ \beta _i$ with $\alpha _i$ and that intertwines the $\mathcal {H}_i$-action with the $\mathcal {G}_i$-action via $\mathcal {J}_i$, for both $i=1,2$.

We depict this as

As an analogue of this in the Hamiltonian setting, we propose the following definitions (more motivation for which will be given in the coming subsections).

Definition 1.43 Let $(\mathcal {G},\Omega )\rightrightarrows M$ be a symplectic groupoid and let $\mathcal {H}\rightrightarrows (S,\omega )$ be a Lie groupoid over a pre-symplectic manifold. We call a Lie groupoid map $\mathcal {J}:\mathcal {H}\to \mathcal {G}$ of Hamiltonian type if

\[ \mathcal{J}^*\Omega=(t_\mathcal{H})^*\omega-(s_\mathcal{H})^*\omega. \]

Definition 1.44 Let $\mathcal {J}_1:\mathcal {H}_1\to \mathcal {G}_1$ and $\mathcal {J}_2:\mathcal {H}_2\to \mathcal {G}_2$ be of Hamiltonian type. By a Hamiltonian Morita equivalence from $\mathcal {J}_1$ to $\mathcal {J}_2$ we mean a Morita equivalence (in the sense of Definition 1.42) with the extra requirement that $(P,\omega _P,\alpha _1,\alpha _2)$ is a symplectic Morita equivalence and that

(46)\begin{equation} j^*\omega_P=(\beta_1)^*\omega_1-(\beta_2)^*\omega_2. \end{equation}

The same type of arguments as for Morita equivalence of Lie and symplectic groupoids (see [Reference XuXu91]) show that Hamiltonian Morita equivalence indeed defines an equivalence relation.

1.5.2 Morita equivalence between groupoid maps of action type

To see that the equivalence relation(s) in the previous subsection induce an equivalence relation between Lie groupoid actions (respectively, Hamiltonian actions), the key remark is that a left action of a Lie groupoid $\mathcal {G}$ along a map $J:S\to M$ gives rise to a map of Lie groupoids covering $J$:

(47)\begin{equation} {\rm pr}_\mathcal{G}:\mathcal{G}\ltimes S\to \mathcal{G}. \end{equation}

Further, notice that the groupoid map (47) is of Hamiltonian type precisely when the action is Hamiltonian (that is, when (5) holds).

In the remainder of this subsection, we further unravel what it means for two Hamiltonian actions to be Morita equivalent. The starting point for this is the following example, which concerns the modules appearing in Theorems 1.33 and 1.36.

Example 1.46 Let $\mathcal {G}_1\rightrightarrows M_1$ be a Lie groupoid acting along $J:S\to M_1$ and suppose that we are given a Morita equivalence $(P,\alpha _1,\alpha _2)$ from $\mathcal {G}_1$ to another Lie groupoid $\mathcal {G}_2\rightrightarrows M_2$. Consider the associated $\mathcal {G}_2$-action along $P_*(J):P\ast _{\mathcal {G}_1}S\to M_2$. The Morita equivalence from $\mathcal {G}_1$ to $\mathcal {G}_2$ extends to a canonical Morita equivalence between these two actions:

Here the upper left action is induced by the diagonal $\mathcal {G}_1$-action, whereas the upper right action is induced by the $\mathcal {G}_2$-action on the first factor. When $(\mathcal {G}_1,\Omega _1)$ and $(\mathcal {G}_2,\Omega _2)$ are symplectic groupoids, the action along $J_1:(S_1,\omega _1)\to M_1$ is Hamiltonian, and the Morita equivalence $(P,\omega _P,\alpha _1,\alpha _2)$ is symplectic, then the associated $(\mathcal {G}_2,\Omega _2)$-action along $P_*(J):(P\ast _{\mathcal {G}_1}S,\omega _{PS})\to M_2$ is Hamiltonian. In this case, the above Morita equivalence is Hamiltonian.

In fact, we will show that more is true.

Here, for convenience, we used the following terminology.

Definition 1.48 Let $\mathcal {J}:\mathcal {H}\to \mathcal {G}$ be a map of Lie groupoids covering $J:S\to M$. We say that $\mathcal {J}$ is of action type if there is a smooth left action of $\mathcal {G}$ along $J$ and an isomorphism of Lie groupoids from $\mathcal {G}\ltimes S$ to $\mathcal {H}$ that covers the identity on $S$ and makes the following diagram commute.

This has the following more insightful characterization.

This is readily verified. To prove Proposition 1.47 we use the following closely related lemma, the proof of which is also left to the reader.

Lemma 1.50 Let $\mathcal {J}_1:\mathcal {H}_1\to \mathcal {G}_1$ and $\mathcal {J}_2:\mathcal {H}_2\to \mathcal {G}_2$ be maps of Lie groupoids and let a Morita equivalence between them (denoted as Definition 1.42) be given. Let $q\in Q$, and denote $p=j(q)$, $p_i=\beta _i(q)$ and $x_i=J_i(p_i)$ for $i=1,2$. Then we have a commutative square

in which all vertical arrows are diffeomorphisms. In particular, $\mathcal {J}_1$ is of action type if and only if $j$ restricts to a diffeomorphism between the $\beta _2$- and $\alpha _2$-fibers. Analogous statements hold for $\mathcal {J}_2$, replacing $\alpha _2$ and $\beta _2$ by $\alpha _1$ and $\beta _1$.

Proof of Proposition 1.47 Suppose that we are given Lie groupoids $\mathcal {G}_1\rightrightarrows M_1$ and $\mathcal {G}_2\rightrightarrows M_2$, together with a $\mathcal {G}_1$-module $J_1:S_1\to M_1$ and a $\mathcal {G}_2$-module $J_2:S_2\to M_2$ and a Morita equivalence between the associated Lie groupoid maps (47), denoted as in Definition 1.42. It follows from Lemma 1.50 that the map

(48)\begin{align} (j,\beta_1):Q\to P\times_{M_1}S_1 \end{align}

is a diffeomorphism. The diagonal action of $\mathcal {G}_1$ along $\alpha _1\circ {\rm pr}_P:P\times _{M_1}S_1\to M_1$ induces an action of $\mathcal {G}_1\ltimes S_1$ along ${\rm pr}_{S_1}:P\times _{M_1}S_1\to S_1$, which is the upper left action in Example 1.46. The diffeomorphism (48) intertwines $\beta _1$ with $\text {pr}_{S_1}$ and is equivariant with respect this action. In particular, by principality of the $\mathcal {G}_1\ltimes S_1$-action, there is an induced diffeomorphism

(49)\begin{equation} S_2\xrightarrow{\sim} P\ast_{\mathcal{G}_1}S_1. \end{equation}

One readily verifies that, when identifying $Q$ with $P\times _{M_1}S_1$ via (48) and $S_2$ with $P\ast _{\mathcal {G}_1}S_1$ via (49), the given Morita equivalence is identified with that in Example 1.46. Furthermore, when $(\mathcal {G}_1,\Omega _1)$ and $(\mathcal {G}_2,\Omega _2)$ are symplectic groupoids, $(S_1,\omega _1)$ and $(S_2,\omega _2)$ are symplectic manifolds, the given actions along $J_1$ and $J_2$ are Hamiltonian and the Morita equivalence between $(\mathcal {G}_1,\Omega _1)$ and $(\mathcal {G}_2,\Omega _2)$ is symplectic, then one readily verifies that (49) is a symplectomorphism from $(S_2,\omega _2)$ to $(P\ast _{\mathcal {G}_1}S_1,\omega _{PS_1})$ if and only if the relation (46) is satisfied. This proves the proposition.

1.5.3 The transverse local model

In this paper we will mainly be interested in Hamiltonian Morita equivalences between Hamiltonian actions, rather than between the more general groupoid maps of Hamiltonian type (as in Definition 1.43). There is, however, one important exception to this.

Example 1.51 This example gives a Hamiltonian Morita equivalence between the local model for Hamiltonian actions and a groupoid map $\mathcal {J}_\mathfrak {p}$ that is built out of less data and is often easier to work with. The use of this Morita equivalence makes many of the proofs in § 2.2 both simpler and more conceptual. Let $(\mathcal {G}_\theta,\Omega _\theta )$ be the symplectic groupoid (30) and let $J_\theta :(S_\theta,\omega _{S_\theta })\to M_\theta$ be the Hamiltonian $(\mathcal {G}_\theta,\Omega _\theta )$-space (32). The Morita equivalence of Example 1.30 extends to a Hamiltonian Morita equivalence between the action along $J_\theta$ and a groupoid map of Hamiltonian type from $H\ltimes (\mathfrak {h}^0\oplus V)$ to $G\ltimes \mathfrak {g}^*$ (restricted to appropriate opens). To see this, let $\mathfrak {p}:\mathfrak {h}^*\to \mathfrak {g}^*$ be an $H$-equivariant splitting of (33). Consider the $H$-equivariant map

(50)\begin{equation} J_\mathfrak{p}:\mathfrak{h}^0\oplus V\to \mathfrak{g}^*, \quad (\alpha,v)\mapsto \alpha+\mathfrak{p}(J_V(v)), \end{equation}

where $J_V:V\to \mathfrak {h}^*$ is the quadratic momentum map (21). By $H$-equivariance, this lifts to a groupoid map

(51)\begin{equation} \mathcal{J}_\mathfrak{p}:H\ltimes (\mathfrak{h}^0\oplus V)\to G\ltimes \mathfrak{g}^*,\quad (h,\alpha,v)\mapsto (h,J_\mathfrak{p}(\alpha,v)). \end{equation}

This groupoid map is not of action type, but it is of Hamiltonian type with respect to the pre-symplectic form $0\oplus \omega _V$ on $\mathfrak {h}^0\oplus V$ and there is a canonical Hamiltonian Morita equivalence

that relates the central orbit in $S_\theta$ to the origin in $\mathfrak {h}^0\oplus V$. Here $W_\theta :={\rm pr}_\mathfrak {g}^*(\Sigma _\theta )$ and $U_\theta :=J_\mathfrak {p}^{-1}(W_\theta )$ are invariant open neighbourhoods of the respective origins in $\mathfrak {g}^*$ and $\mathfrak {h}^0\oplus V$. Furthermore, the map $\beta _\mathfrak {p}$ is defined as

\[ \beta_\mathfrak{p}: \Sigma_{\theta _{{\rm pr}_{\mathfrak{h}^*}}}{\times}{_{J_V}} V\to U_{\theta},\quad (p,\alpha,v)\mapsto (\alpha-\mathfrak{p}(J_V(v)),v). \]

With this in mind, we think of the groupoid map $\mathcal {J}_\mathfrak {p}$ as a local model for the ‘transverse part’ of a Hamiltonian action near a given orbit.

1.5.4 Elementary Morita invariants

As will be apparent in the rest of this paper, many invariants for Morita equivalence between Lie groupoids have analogues for Morita equivalence between Hamiltonian actions: in fact, the canonical Hamiltonian stratification can be thought of as an analogue of the canonical stratification on the leaf space of a proper Lie groupoid. In this subsection we give analogues of Proposition 1.26. We start with a version for Lie groupoid maps.

The proof is straightforward.

Example 1.53 The Morita equivalence in Example 1.51 induces an identification (of maps of topological spaces) between the transverse momentum map $\underline {J_\theta }$ and (a restriction of) the map

\[ \underline{J_\mathfrak{p}}:(\mathfrak{h}^0\oplus V)/H\to \mathfrak{g}^*/G. \]

We now turn to Morita equivalences between Hamiltonian actions.

Proof. Part (a) is immediate from Proposition 1.52(b). For the proof of part (b) observe that, by Proposition 1.52(c), the isomorphism $\varphi _q$ restricts to one between ${\rm Ker}(\underline {{\rm d} J}_1)_{p_1}$ and ${\rm Ker}(\underline {{\rm d} J}_2)_{p_2}$, so that we obtain an isomorphism of representations, compatible with part (a), and given by

(52)\begin{equation} \mathcal{S}\mathcal{N}_{p_2}\to \mathcal{S}\mathcal{N}_{p_1},\quad [v]\mapsto [{\rm d} \beta_1(\hat{v})], \end{equation}

where $\hat {v}\in T_qQ$ is any vector such that ${\rm d} \beta _2(\hat {v})=v$ and ${\rm d} j(\hat {v})=0$. Note here (to see that such $\hat {v}$ exists) that, given $v\in {\rm Ker}({\rm d} J_2)_{p_2}$ and $\hat {w}\in T_qQ$ such that ${\rm d} \beta _2(\hat {w})=v$, we have ${\rm d} j(\hat {w})\in {\rm Ker}({\rm d}\alpha _2)$, hence by Lemma 1.50 there is a $\hat {u}\in {\rm Ker}({\rm d}\beta _2)_q$ such that ${\rm d} j(\hat {u})={\rm d} j(\hat {w})$, so that $\hat {v}:=\hat {w}-\hat {u}$ has the desired properties. With this description of (52) it is immediate from (46) that (52) pulls $(\omega _1)_{p_1}$ back to $(\omega _2)_{p_2}$, which concludes the proof.

We can now give a more conceptual proof of Lemma 1.41.

For Hamiltonian Morita equivalences as in Example 1.51 (where one of the two groupoid maps is not of action type) it is not clear to us whether there is a satisfactory generalization of Proposition 1.54. The arguments in the proof of that proposition do show the following, which will be enough for our purposes.

Proposition 1.55 Let a Hamiltonian Morita equivalence (denoted as in Definitions 1.42 and 1.44) between groupoids maps $\mathcal {J}_1:\mathcal {H}_1\to (\mathcal {G}_1,\Omega _1)$ and $\mathcal {J}_2:\mathcal {H}_2\to (\mathcal {G}_2,\Omega _2)$ of Hamiltonian type be given. Suppose that $p_1\in S_1$ and $p_2\in S_2$ belong to $Q$-related orbits and let $q\in Q$ such that $\beta _1(q)=p_1$ and $\beta _2(q)=p_2$. Further, assume that $\mathcal {J}_1$ is of action type and the canonical injection

\[ \mathcal{S}\mathcal{N}_{p_2}:=\frac{{\rm Ker}({d} J_2)_{p_2}}{{\rm Ker}({d} J_2)_{p_2}\cap T_{p_2}\mathcal{O}}\hookrightarrow {\rm Ker}(\underline{{d} J}_2)_{p_2} \]

is an isomorphism. The form $\omega _2$ on the base $S_2$ of $\mathcal {H}_2$ may be degenerate. Then:

1. (a) the form $\omega _2$ descends to a linear symplectic form $(\omega _2)_{p_2}$ on $\mathcal {S}\mathcal {N}_{p_2}$, which is invariant under the $(\mathcal {H}_2)_{p_2}$-action defined by declaring the isomorphism with ${\rm Ker}(\underline {{d} J})_{p_2}$ to be equivariant;

2. (b) there is a symplectic linear isomorphism $(\mathcal {S}\mathcal {N}_{p_1},\omega _{p_1})\cong (\mathcal {S}\mathcal {N}_{p_2},\omega _{p_2})$ that is compatible with the isomorphism of Lie groups $\Phi _q:(\mathcal {H}_1)_{p_1}\xrightarrow {\sim }(\mathcal {H}_2)_{p_2}$.

2.1.1 Stratifications of topological spaces

In this paper, by a stratification we mean the following.

Due to the connectedness assumption on the strata, the frontier condition (a priori of a global nature) can be verified locally with the following lemma.

The constructions of the stratifications in this paper follow a general pattern: one first defines a partition $\mathcal {P}$ of $X$ into manifolds (possibly disconnected, with connected components of varying, but bounded, dimension) which in a local model for $X$ have a particularly simple description. This partition $\mathcal {P}$ is often natural to the given geometric situation from which $X$ arises. Then, one passes to the partition $\mathcal {S}:=\mathcal {P}^{\rm c}$ consisting of the connected components of the members of $\mathcal {P}$, and verifies that $\mathcal {S}$ is a stratification of $X$.

Example 2.5 The leaf space of a proper Lie groupoid admits a canonical stratification. To elaborate, let $\mathcal {G}\rightrightarrows M$ be a proper Lie groupoid, meaning that $\mathcal {G}$ is Hausdorff and the map $(t,s):\mathcal {G}\to M\times M$ is proper. This is equivalent to requiring that $\mathcal {G}$ is proper at every $x\in M$ (as in Definition 1.7) and that its leaf space $\underline {M}$ is Hausdorff [Reference del HoyodHo13, Proposition 5.1.3]. In fact, $\underline {M}$ is locally compact, second countable and Hausdorff (so, in particular, it is paracompact). To define the stratifications of $M$ and $\underline {M}$, first consider the partition $\mathcal {P}_\mathcal {M}(M)$ of $M$ by Morita types. This is given by the equivalence relation: $x_1\sim _\mathcal {M} x_2$ if and only if there are invariant opens $V_1$ and $V_2$ around $\mathcal {L}_{x_1}$ and $\mathcal {L}_{x_2}$, respectively, together with a Morita equivalence

\[ \mathcal{G}\vert_{V_1}\simeq \mathcal{G}\vert_{V_2}, \]

that relates $\mathcal {L}_{x_1}$ to $\mathcal {L}_{x_2}$. Its members are invariant and therefore descend to a partition $\mathcal {P}_\mathcal {M}(\underline {M})$ of the leaf space $\underline {M}$. The partitions $\mathcal {S}_{\rm Gp}(M)$ and $\mathcal {S}_{\rm Gp}(\underline {M})$ obtained from $\mathcal {P}_\mathcal {M}(M)$ and $\mathcal {P}_\mathcal {M}(\underline {M})$ after passing to connected components form the so-called canonical stratifications of the base $M$ and the leaf space $\underline {M}$ of the Lie groupoid $\mathcal {G}$. These indeed form stratifications. This is proved in [Reference Pflaum, Posthuma and TangPPT14] and [Reference Crainic and MestreCM17], using the local description given by the linearization theorem for proper Lie groupoids (see [Reference WeinsteinWei02, Reference ZungZun06, Reference Crainic and StruchinerCS13, Reference Fernandes and del HoyoFdH18]). There, the partition by Morita types is defined by declaring that $x,y\in M$ belong to the same Morita type if and only if there is an isomorphism of Lie groups

\[ \mathcal{G}_x\cong \mathcal{G}_y \]

together with a compatible linear isomorphism

\[ \mathcal{N}_x\cong \mathcal{N}_y \]

between the normal representations of $\mathcal {G}$ at $x$ and $y$, as in (10). This is equivalent to the description given before, as a consequence of Proposition 1.26(b) and the linearization theorem.

Often there are various different partitions that, after passing to connected components, induce the same stratification. This too can be checked locally, using the following lemma.

Lemma 2.6 Let $\mathcal {P}_1$ and $\mathcal {P}_2$ be partitions of a topological space $X$ into manifolds (equipped with the subspace topology) with connected components of possibly varying dimension. Then the partitions $\mathcal {P}_1^c$ and $\mathcal {P}_2^c$, obtained after passing to connected components, coincide if and only if every $x\in X$ admits an open neighbourhood $U$ in $X$ such that

\[ P_1\cap U=P_2 \cap U, \]

where $P_1$ and $P_2$ are the members of $\mathcal {P}_1$ and $\mathcal {P}_2$ through $x$.